0200 Hamiltonian Circuits

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0200 Hamiltonian Circuits

• Review Day

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• Hamiltonian Circuits
• 3 Rules
• Start and stop at the same vertex
• Touch every vertex
• Pass through every vertex only once
• Example

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Fundamental principle of counting If there are
a ways of choosing on thing, b ways of choosing a
second after the first is chosen, and z ways of
choosing the last item after the earlier choices,
then the total number of choice patterns is a
X b X c X . X z
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FPC for Hamiltonian Circuits Given a complete
graph with n vertices, once you choose the
beginning vertex, there are n-1 ways to choose
the second vertex, the next will have n-2 ways to
be chosen, and so on... So a complete
Hamiltonian Circuit will have (n-1)! POSSIBLE
routes. Since a route is the same forwards and
backwards, there are (n-1)!/2 DIFFERENT routes.
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• The Traveling Salesman Problem adds weights
  based on the context of the problem to each of
  the edges. To solve this problem and find a
  minimum-cost circuit we can use one of three
  different methods
• Brute force method will give ALL possible
  routes, therefore ALWAYS finding the absolute
  minimum-cost circuit
• Nearest-neighbor algorithm will give A
  minimum-cost circuit, but maybe NOT the absolute
  smallest and may depend on the starting vertex
• Sorted-edges algorithm will give A minimum-cost
  circuit, but maybe NOT the absolute smallest OR
  the same as the nearest-neighbor

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• Brute Force Method
• Find all possible routes using the method of
  trees
• Calculate all the routes costs
• Pick the smallest number

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• Nearest-neighbor algorithm In the context that
  the vertices are cities
• You will be given a starting city (if not must
  calculate starting at every city)
• Visit the nearest city
• Visit the nearest city that has not already
  been visited
• Repeat step 3 over and over
• - Return to the starting city only once all
  the cities have been visited
• 5. Tally the weights of the edges used

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• Sorted-edges Algorithm Not dependent on a
  starting vertex
• Remove all the edges
• Sort the edges from smallest to largest
• Place the smallest edge back on the graph
• Place the next smallest edge back on the graph
• Place the third smallest edge and so on until a
  HC is created
• - Skip edges that create a circuit on the
  graph that doesnt include all the vertices
• - Skip edges that make the valence of a
  vertex more than 2
• 6.Tally the weights of the edges used

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A Hamiltonian Circuit must use EVERY vertex on a
given graph A Spanning Tree must also use EVERY
vertex on a graph, BUT in a spanning tree closing
the circuit is not necessary and its perfectly
ok to retrace your steps
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• Kruskals Algorithm only algorithm we know to
  find a minimum-cost spanning tree
• Remove all the edges
• Sort the edges from smallest to largest
• Place the smallest edge on the graph
• Place the second smallest edge on the graph
• Continue placing edges on the graph until the
  graph is connected (Can I trace my pencil over
  the entire graph without lifting my pencil?)
• Locate any circuits and remove the largest edge

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Can you give an example why we would use a
spanning tree and not a Hamiltonian Circuit?
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Order-requirement digraph the longest path is
the shortest possible completion time, known as
the critical path
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64 from book At a large toy store, scooters
arrive unassembled in boxes. To assemble a
scooter, the following tasks must be
performed TASK 1 Remove parts from the
box TASK 2 Attach wheels to the footboard TASK
3 Attach vertical housing TASK 4 Attach
handlebars to vertical housing TASK 5 Put on
reflector tape TASK 6 Attach bell to
handlebars TASK 7 Attach decals TASK 8 Attach
kickstand TASK 9 Attach safety instructions to
handlebars Its like a puzzle!!
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HAPPY STUDYING!!!